Metamath Proof Explorer

Theorem tfsconcatrnss12

Description: The range of the concatenation of transfinite sequences is a superset of the ranges of both sequences. Theorem 3 in Grzegorz Bancerek, "Epsilon Numbers and Cantor Normal Form", Formalized Mathematics, Vol. 17, No. 4, Pages 249–256, 2009. DOI: 10.2478/v10037-009-0032-8 (Contributed by RP, 2-Mar-2025)

Ref Expression
Hypothesis tfsconcat.op 
|- .+ = ( a e. _V , b e. _V |-> ( a u. { <. x , y >. | ( x e. ( ( dom a +o dom b ) \ dom a ) /\ E. z e. dom b ( x = ( dom a +o z ) /\ y = ( b ` z ) ) ) } ) )
Assertion tfsconcatrnss12 
|- ( ( ( A Fn C /\ B Fn D ) /\ ( C e. On /\ D e. On ) ) -> ( ran A C_ ran ( A .+ B ) /\ ran B C_ ran ( A .+ B ) ) )

Proof

Step Hyp Ref Expression
1 tfsconcat.op 
 |-  .+ = ( a e. _V , b e. _V |-> ( a u. { <. x , y >. | ( x e. ( ( dom a +o dom b ) \ dom a ) /\ E. z e. dom b ( x = ( dom a +o z ) /\ y = ( b ` z ) ) ) } ) )
2 1 tfsconcatrn 
 |-  ( ( ( A Fn C /\ B Fn D ) /\ ( C e. On /\ D e. On ) ) -> ran ( A .+ B ) = ( ran A u. ran B ) )
3 ssun1 
 |-  ran A C_ ( ran A u. ran B )
4 id 
 |-  ( ran ( A .+ B ) = ( ran A u. ran B ) -> ran ( A .+ B ) = ( ran A u. ran B ) )
5 3 4 sseqtrrid 
 |-  ( ran ( A .+ B ) = ( ran A u. ran B ) -> ran A C_ ran ( A .+ B ) )
6 ssun2 
 |-  ran B C_ ( ran A u. ran B )
7 6 4 sseqtrrid 
 |-  ( ran ( A .+ B ) = ( ran A u. ran B ) -> ran B C_ ran ( A .+ B ) )
8 5 7 jca 
 |-  ( ran ( A .+ B ) = ( ran A u. ran B ) -> ( ran A C_ ran ( A .+ B ) /\ ran B C_ ran ( A .+ B ) ) )
9 2 8 syl 
 |-  ( ( ( A Fn C /\ B Fn D ) /\ ( C e. On /\ D e. On ) ) -> ( ran A C_ ran ( A .+ B ) /\ ran B C_ ran ( A .+ B ) ) )